I came up with the following problem:
Let $\Pi_A$ and $\Pi_B$ be two projection operators on two disjoint subspaces of a certain Hilbert space $\mathcal H$ and let $\rho$ be unit trace, positive, hermitian matrix operating on $\mathcal H$.
Show that $Tr((\Pi_A+\Pi_B)\rho) = Tr((\Pi_A)\rho) + Tr((\Pi_B)\rho)$ whenever $\rho$ can be diagonalized in the same basis as $\Pi_A$ and $\Pi_B$.
Then, as far as I understand it follows by the linearity of the trace that the equality always holds even if $\rho$ can not be diagonalized in the same basis as the projection operators. Is that correct?